Teaching measurement

1. Teaching measurement MATH 124

2. Experiences with measurement • What were your experiences with measurement in elementary school? Did you have any out-of-school measurement experiences? What has been more helpful in learning measurement?

3. Key ideas in teaching measurement • Making comparisons between what is being measured and some suitable standard of measure (Burns) • It is key that children get ample practice with measurement • It is also key to understand that measurements are never exact, and that children need to learn what it means to be “close enough,” and to make estimates

4. What is measurement? • The term measurement may refer to the process of finding the value of a quantity, or may refer to the result of that process. • The process of direct measurement of a characteristic involves matching that characteristic of the object by using (usually repeatedly) a unit, which is a different object having the same characteristic. Indirect measurement involves mathematical or scientific principles (e.g. direct measurement of weight via pan balance; indirect measurement via bathroom scale).

5. What should be taught in the elementary curriculum? • Making comparisons between objects by matching, without use of measuring tools. For example, students compare who is taller or who has a longer foot. • Comparing objects with nonstandard units. For example, they can measure height using parts of the body, straws, books, etc. • Comparing objects with standard units, both metric and standard. • Choosing suitable units for specific measurements

6. The four stages described above should be used in teaching length, time, weight/mass, volume/capacity, temperature and area • Before third grade, students should use direct comparisons and nonstandard units to make comparisons; after third grade, they can move into standard units of measure (though this is not quite what the CCSSM say)

7. Example: body ratios • Estimate how many times a piece of string equal to a student’s height would wrap around their head as a headband, then measure (the ratio should be close to 1:3) • Estimate, then find the ratio between the length of a student’s foot length and height (should be close to 1:6) • “Are you a square or a rectangle?” • “Fit a drawing of yourself on a 5x8 index card” • Note: caution should be used when working with student body measurements

8. Common Core: Kindergarten • CCSS.Math.Content.K.MD.A.1Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. • CCSS.Math.Content.K.MD.A.2Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.

9. Common Core: 1st grade • CCSS.Math.Content.1.MD.A.1Order three objects by length; compare the lengths of two objects indirectly by using a third object. • CCSS.Math.Content.1.MD.A.2Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps. • CCSS.Math.Content.1.MD.B.3Tell and write time in hours and half-hours using analog and digital clocks.

10. Common Core: 2nd grade • CCSS.Math.Content.2.MD.A.1Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. • CCSS.Math.Content.2.MD.A.2Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. • CCSS.Math.Content.2.MD.A.3Estimate lengths using units of inches, feet, centimeters, and meters.

11. CCSS.Math.Content.2.MD.A.4Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. • CCSS.Math.Content.2.MD.B.5Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem. • CCSS.Math.Content.2.MD.B.6Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

12. CCSS.Math.Content.2.MD.C.7Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. • CCSS.Math.Content.2.MD.C.8Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

13. Common Core: 3rd grade • CCSS.Math.Content.3.MD.A.1Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. • CCSS.Math.Content.3.MD.A.2Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).1 Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.

14. CCSS.Math.Content.3.MD.C.5Recognize area as an attribute of plane figures and understand concepts of area measurement. • CCSS.Math.Content.3.MD.C.5.aA square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area. • CCSS.Math.Content.3.MD.C.5.bA plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. • CCSS.Math.Content.3.MD.C.6Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). • CCSS.Math.Content.3.MD.C.7Relate area to the operations of multiplication and addition. • CCSS.Math.Content.3.MD.C.7.aFind the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

15. CCSS.Math.Content.3.MD.C.7.bMultiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. • CCSS.Math.Content.3.MD.C.7.cUse tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. • CCSS.Math.Content.3.MD.C.7.dRecognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. • CCSS.Math.Content.3.MD.D.8Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

16. Common Core: 4th grade • CCSS.Math.Content.4.MD.A.1Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), … • CCSS.Math.Content.4.MD.A.2Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.

17. CCSS.Math.Content.4.MD.A.3Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. • CCSS.Math.Content.4.MD.C.5Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: • CCSS.Math.Content.4.MD.C.5.aAn angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.

18. CCSS.Math.Content.4.MD.C.5.bAn angle that turns through n one-degree angles is said to have an angle measure of n degrees. • CCSS.Math.Content.4.MD.C.6Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. • CCSS.Math.Content.4.MD.C.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

19. Common Core: 5th grade • CCSS.Math.Content.5.MD.A.1Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

20. Metric units • Cuisenaire rods • Which things are as long as the white rod? • Which things are as long as an orange rod? • Which things are as long as 10 orange rods? • Metric body measurements: what is 1cm long; 1 dm long; 1 m long? • Students make their own metric ruler

21. Metric sense • If an elevator can hold ten average eight graders, is the weight limit • 8 kg, 80 kg or 800 kg? • You could probably eat • 10 g, 500 g, 1000 g of pancakes • and drink • 10 mL, 500 mL, 1000 mL of juice for breakfast.

22. Lengths and angles • Different words are used as synonyms for length: height, width, and even perimeter and circumference. • Question: How do we find lengths of “crooked” lines? • How do we compare sizes of angles, i.e. how do we determine which angle is larger? What are some mistakes that children may make when comparing angles? How do we measure angles smaller than 1 degree? (Look on page 543.)

23. Perimeter • Perimeter is the distance around a two-dimensional figure. We assume the figure is closed. A special case is when the figure is a polygon: then the perimeter is equal to the sum of the lengths of all the sides.

24. Area • Area is the amount of space a two-dimensional region occupies. The figure again needs to be closed. • The word area can refer both to an attribute and measurement. • Area of a region is the number of area units (typically squares) required to cover the region. In the case of a rectangle with sides with integer side lengths, say l and w, the rectangle can be covered by a grid of w rows and l columns, for a total of lw unit squares. This formula can be generalized to any rectangle.

25. Perimeter and area • Perimeter and area are not related. Bigger area may mean bigger, smaller, or equal perimeter.

26. Area • Discussion 2 on page 547: what are appropriate units for measuring area? • Activity 1 on page 548: how are square units related? How many square inches are in a square foot? How many square feet in a square yard? How many square centimeters in a square meter?